After the Important Visitor mission, you'll have the option to call Ilyana. Do so to unlock her date mission, Make Ilyana Sweat. Once that's over with, a new favor will appear in K-Bar listed as "Real men don't karaoke." You'll have to intentionally do bad at a song in order to pass this favor. I recommend muting your TV.
After completing this favor, every karaoke song is unlocked. Go through each one and score over 90% on each. If you're having trouble with the halfway notes, practice just before a song or on an earlier, easier song to find the threshold.
Bonus achievement for resisting the temptation to sing along with Take on Me.